List of integrals of hyperbolic functions

2025-03-13T04:36:49Z

2601:410:4200:7010:E050:35B3:3CC2:6530:

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The following is a list of [[integral]]s ([[anti-derivative]] functions) of [[hyperbolic function]]s. For a complete list of integral functions, see [[list of integrals]].

In all formulas the constant ''a'' is assumed to be nonzero, and ''C''
denotes the [[constant of integration]].

==Integrals involving only hyperbolic sine functions==

{{startplainlist|indent=1}}
* <math>\int\sinh ax\,dx = \frac{1}{a}\cosh ax+C</math>
* <math>\int\sinh^2 ax\,dx = \frac{1}{4a}\sinh 2ax - \frac{x}{2}+C</math>
* <math>\int\sinh^n ax\,dx = \begin{cases}
\frac{1}{an}(\sinh^{n-1} ax)(\cosh ax) - \frac{n-1}{n}\displaystyle\int\sinh^{n-2} ax\,dx, & n>0 \\
\frac{1}{a(n+1)}(\sinh^{n+1} ax)(\cosh ax) - \frac{n+2}{n+1}\displaystyle\int\sinh^{n+2}ax\,dx, & n<0, n\neq -1
\end{cases}
</math>
* <math>\begin{align}
\int\frac{dx}{\sinh ax} &= \frac{1}{a} \ln\left|\tanh\frac{ax}{2}\right|+C \\
&= \frac{1}{a} \ln\left|\frac{\cosh ax - 1}{\sinh ax}\right|+C \\
&= \frac{1}{a} \ln\left|\frac{\sinh ax}{\cosh ax + 1}\right|+C \\
&= \frac{1}{2a} \ln\left|\frac{\cosh ax - 1}{\cosh ax + 1}\right|+C
\end{align}</math>
* <math>\int\frac{dx}{\sinh^n ax} = -\frac{\cosh ax}{a(n-1)\sinh^{n-1} ax}-\frac{n-2}{n-1}\int\frac{dx}{\sinh^{n-2} ax} \qquad\mbox{(for }n\neq 1\mbox{)}</math>
* <math>\int x\sinh ax\,dx = \frac{1}{a} x\cosh ax - \frac{1}{a^2}\sinh ax+C</math>
* <math>\int (\sinh ax)(\sinh bx)\,dx = \frac{1}{a^2-b^2} \big(a(\sinh bx)(\cosh ax) - b(\cosh bx)(\sinh ax)\big)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}</math>
{{endplainlist}}

==Integrals involving only hyperbolic cosine functions==

{{startplainlist|indent=1}}
* <math>\int\cosh ax\,dx = \frac{1}{a}\sinh ax+C</math>
* <math>\int\cosh^2 ax\,dx = \frac{1}{4a}\sinh 2ax + \frac{x}{2}+C</math>
* <math>\int\cosh^n ax\,dx = \begin{cases}
\frac{1}{an}(\sinh ax)(\cosh^{n-1} ax) + \frac{n-1}{n}\displaystyle\int\cosh^{n-2} ax\,dx, & n>0 \\
-\frac{1}{a(n+1)}(\sinh ax)(\cosh^{n+1} ax) + \frac{n+2}{n+1}\displaystyle\int\cosh^{n+2}ax\,dx, & n<0, n\neq -1
\end{cases}
</math>
* <math>\begin{align}
\int\frac{dx}{\cosh ax} &= \frac{2}{a} \arctan e^{ax}+C \\
&= \frac{1}{a} \arctan (\sinh ax)+C
\end{align}</math>
* <math>\int\frac{dx}{\cosh^n ax} = \frac{\sinh ax}{a(n-1)\cosh^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\cosh^{n-2} ax} \qquad\mbox{(for }n\neq 1\mbox{)}</math>
* <math>\int x\cosh ax\,dx = \frac{1}{a} x\sinh ax - \frac{1}{a^2}\cosh ax+C</math>
* <math>\int x^2 \cosh ax\,dx = -\frac{2x \cosh ax}{a^2} + \left(\frac{x^2}{a}+\frac{2}{a^3}\right) \sinh ax+C</math>
* <math>\int (\cosh ax)(\cosh bx)\,dx = \frac{1}{a^2-b^2} \big(a(\sinh ax)(\cosh bx) - b(\sinh bx)(\cosh ax)\big)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}</math>
*<math>\int \frac{dx}{1+\cosh(ax)} = \frac{2}{a} \frac{1}{1+e^{-ax}}+C\quad</math> or <math>\frac{2}{a}</math> times [[Logistic function|The Logistic Function]]
{{endplainlist}}

==Other integrals==

===Integrals of hyperbolic tangent, cotangent, secant, cosecant functions===

{{startplainlist|indent=1}}
* <math>\int \tanh x \, dx = \ln \cosh x + C</math>
* <math>\int\tanh^2 ax\,dx = x - \frac{\tanh ax}{a}+C</math>
* <math>\int \tanh^n ax\,dx = -\frac{1}{a(n-1)}\tanh^{n-1} ax+\int\tanh^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}</math>
* <math>\int \coth x \, dx = \ln| \sinh x | + C , \text{ for } x \neq 0 </math>
* <math>\int \coth^n ax\,dx = -\frac{1}{a(n-1)}\coth^{n-1} ax+\int\coth^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}</math>
* <math>\int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C</math>
* <math>\int \operatorname{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C = \ln\left|\coth{x}-\operatorname{csch}{x}\right|+C, \text{ for } x \neq 0 </math>
{{endplainlist}}

===Integrals involving hyperbolic sine and cosine functions===

{{startplainlist|indent=1}}
* <math>\int (\cosh ax)(\sinh bx)\,dx = \frac{1}{a^2-b^2} \big(a(\sinh ax)(\sinh bx) - b(\cosh ax)(\cosh bx)\big)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}</math>
* <math>\begin{align}
\int\frac{\cosh^n ax}{\sinh^m ax}\,dx &= \frac{\cosh^{n-1} ax}{a(n-m)\sinh^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cosh^{n-2} ax}{\sinh^m ax}\,dx \qquad\mbox{(for }m\neq n\mbox{)} \\
&= -\frac{\cosh^{n+1} ax}{a(m-1)\sinh^{m-1} ax} + \frac{n-m+2}{m-1}\int\frac{\cosh^n ax}{\sinh^{m-2} ax}\,dx \qquad\mbox{(for }m\neq 1\mbox{)} \\
&= -\frac{\cosh^{n-1} ax}{a(m-1)\sinh^{m-1} ax} + \frac{n-1}{m-1}\int\frac{\cosh^{n-2} ax}{\sinh^{m-2} ax}\,dx \qquad\mbox{(for }m\neq 1\mbox{)}
\end{align}</math>
* <math>\begin{align}
\int\frac{\sinh^m ax}{\cosh^n ax}\,dx &= \frac{\sinh^{m-1} ax}{a(m-n)\cosh^{n-1} ax} + \frac{m-1}{n-m}\int\frac{\sinh^{m-2} ax}{\cosh^n ax}\,dx \qquad\mbox{(for }m\neq n\mbox{)} \\
&= \frac{\sinh^{m+1} ax}{a(n-1)\cosh^{n-1} ax} + \frac{m-n+2}{n-1}\int\frac{\sinh^m ax}{\cosh^{n-2} ax}\,dx \qquad\mbox{(for }n\neq 1\mbox{)} \\
&= -\frac{\sinh^{m-1} ax}{a(n-1)\cosh^{n-1} ax} + \frac{m-1}{n-1}\int\frac{\sinh^{m -2} ax}{\cosh^{n-2} ax}\,dx \qquad\mbox{(for }n\neq 1\mbox{)}
\end{align}</math>
{{endplainlist}}

===Integrals involving hyperbolic and trigonometric functions===

{{startplainlist|indent=1}}
* <math>\int \sinh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+C</math>
* <math>\int \sinh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\sinh(ax+b)\sin(cx+d)+C</math>
* <math>\int \cosh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+C</math>
* <math>\int \cosh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\cosh(ax+b)\sin(cx+d)+C</math>
{{endplainlist}}

{{Lists of integrals}}

{{DEFAULTSORT:Integrals of hyperbolic functions}}
[[Category:Exponentials]]
[[Category:Lists of integrals|Hyperbolic functions]]

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List of integrals of hyperbolic functions